Park City Mathematics Institute Secondary School Teachers Program Summer 2011 Week 2: Monday Zome Building

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Ning: PCMI 2011

Forms

The large Zometool construction for 2011 is a compound of a 120-cell and a Gosset polytope projected into three dimensions. The 120-cell is a four dimensional analog of a dodecahedron, a longtime PCMI favorite [http://mathforum.org/pcmi/truncated120.html]. The Gosset polytope is an eight dimension structure important in geometry and algebra [http://vzome.com/gosset/].

Why combine these two things, which don't even live in the same dimension? You may know that the dodecahedron and icosahedron are duals: put a vertex in each face of one polyhedron, connect adjacent vertices, and you get the other. A Zome creation showing concentric dodecahedron and icosahedron of roughly the same diameter is an inspiration for what we made.

The 120-cell is dual to a four dimensional polytope known as the 600 cell. A projection of the 600 cell into three dimensions can be built in the Zome system, but the model is dense to the point of obscuring structure. The Gosset polytope, however, can be built from two copies of the 600-cell and has a "nicer" projection into three dimensions. The large compound shape shows traits of both structures, e.g., the blue tunnels of the 120-cell and the parallel planes in the Gosset polytope, while illustrating duality. In fact, the compound dodecahedron and icosahedron structure appears three times in layers of our primary 2011 giant Zome build, created under the guidance and enthusiasm of ZomeTool "chief visionary officer" Paul Hildebrandt.

--Brian Hopkins

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